# Markov kernel

In probability theory, a Markov kernel (or stochastic kernel) is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.[1][2]

## Formal definition

(i.e. It associates to each point ${\displaystyle x\in X}$ a probability measure ${\displaystyle \kappa (x,.)}$ on ${\displaystyle (Y,{\mathcal {B}})}$ such that, for every measurable set ${\displaystyle B\in {\mathcal {B}}}$, the map ${\displaystyle x\mapsto \kappa (x,B)}$ is measurable with respect to the ${\displaystyle \sigma }$-algebra ${\displaystyle {\mathcal {A}}}$.)

## Examples

${\displaystyle \kappa (x,B)={\frac {1}{2}}\delta _{x-1}(B)+{\frac {1}{2}}\delta _{x+1}(B),\quad \forall x\in \mathbb {Z} \forall B\in {\mathcal {P}}(\mathbb {Z} )}$,

describes the transition rule for the random walk on ${\displaystyle \mathbb {Z} }$.

${\displaystyle \kappa (x,B)={\begin{cases}\delta _{0}(B)&\quad x=0,\\P[\xi _{1}+\dots +\xi _{x}\in B]&\quad {\text{else,}}\\\end{cases}}}$
${\displaystyle \kappa (i,B)=\Sigma _{j\in B}K_{ij}\quad \forall i\in X\forall B\in {\mathcal {B}}}$.

## Properties

### Semidirect product

${\displaystyle Q(A\times B)=\int _{A}\kappa (x,B)dP(x),\quad \forall A\in {\mathcal {A}}\forall B\in {\mathcal {B}}}$.

### Regular conditional distribution

${\displaystyle P[X\in B|{\mathcal {G}}]=E[\mathbf {1} _{\{X\in B\}}|{\mathcal {G}}]=\kappa (\omega ,B),\quad P-a.s.\forall B\in {\mathcal {G}}}$.

It is called regular conditional distribution of ${\displaystyle X}$ given ${\displaystyle {\mathcal {G}}}$ and is not uniquely defined.

## Estimation

The kernel can be estimated using kernel density estimation.[3]

## References

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§36. Kernels and semigroups of kernels